Profinite groups are another important class of topological groups;
they arise, for example, in infinite Galois theory.

Subgroups, quotients and products

Every subgroup (http://planetmath.org/Subgroup) of a topological group
either has empty interior or is clopen.
In particular, all proper subgroups of a connected topological group
have empty interior.
The closure of any subgroup is also a subgroup,
and the closure of a normal subgroup is normal
(for proofs, see the entry
“closure of sets closed under a finitary operation (http://planetmath.org/ClosureOfSetsClosedUnderAFinitaryOperation)”).
A subgroup of a topological group is itself a topological group,
with the subspace topology.

If G is a topological group and N is a normal subgroup of G,
then the quotient groupG/N is also a topological group,
with the quotient topology.
This quotient G/N is Hausdorff if and only if N is a closed subset of G.

If (Gi)i∈I is a family of topological groups,
then the unrestricted direct product∏i∈IGi
is also a topological group, with the product topology.

Morphisms

The function f is said to be a homomorphism of topological groups
if it is a group homomorphism and is also continuous.
It is said to be an isomorphism of topological groups
if it is both a group isomorphism and a homeomorphism.

Note that it is possible for f to be a continuous groupisomorphism
(that is, a bijective homomorphism of topological groups)
and yet not be an isomorphism of topological groups.
This occurs, for example, if G is ℝ with the discrete topology,
and H is ℝ with its usual topology,
and f is the identity map on ℝ.

Topological properties

While every group can be made into a topological group,
the same cannot be said of every topological space.
In this section we mention some of the properties
that the underlying topological space must have.

Every topological group is bihomogeneous
and completely regular (http://planetmath.org/Tychonoff).
Note that our earlier claim that a topological group
is Hausdorff if and only if its trivial subgroup is closed
follows from this:
if the trivial subgroup is closed,
then homogeneity ensures that all singletons are closed,
and so the space is T1 (http://planetmath.org/T1Space),
and being completely regular is therefore Hausdorff.
A topological group is not necessarily http://planetmath.org/node/1530normal, however,
a counterexample being the unrestricted direct product
of uncountably many copies of the discrete groupℤ.

Every topological group is obviously an H-space.
Consequently, the fundamental group of a topological group is abelian.
Note that because topological groups are homogeneous,
the fundamental group does not depend (up to isomorphism)
on the choice of basepoint.

Other notes

Every topological group possesses a natural uniformity,
which induces the topology.
See the entry about the uniformity of a topological group (http://planetmath.org/UniformStructureOfATopologicalGroup).

A locally compact topological group
possesses a natural measure, called the Haar measure.